eXtended Pom-Pom Model Research Articles

Exact Solution for Viscoelastic Flow in Pipe and Experimental Validation.

The development of analytical methods for viscoelastic fluid flows is challenging. Currently, this problem has been solved for particular cases of multimode differential rheological equations of media state (Giesekus, the exponential form of Phan-Tien-Tanner, eXtended Pom-Pom). We propose a parametric method that yields solutions without additional assumptions. The method is based on the parametric representation of the unknown velocity functions and the stress tensor components as a function of coordinate. Experimental flow visualization based on the SIV (smoke image velocimetry) method was carried out to confirm the obtained results. Compared to the Giesekus model, the experimental data are best predicted by the eXtended Pom-Pom model.

The role of the Gordon–Schowalter derivative term in the constitutive models—improved flexibility of the modified XPP model

Constitutive models that complete the set of equations describing the flow of polymer melts should respect objective thermodynamics and stability conditions ensuring their validity in the whole range of possible deformation flow. However, in practice, a very good description of flow situations can be achieved with the models not complying with the physical assumptions in all respects. Analogously to the term characterizing yield stress in empirical viscoplastic models, the term represented by the Gordon–Schowalter (GS) derivative in the differential constitutive models contributes to better fitting the experimental data, especially shear thinning. Efficiency of the recently presented modified eXtended Pom-Pom model (just one non-linear parameter per mode) implementing the GS derivative term (one additional non-affine motion parameter per mode) is improved (documented on LDPE, HDPE, and polyvinyl butyral (PVB) materials), and a comparison with the exponential Phan-Tien–Tanner (PTT) and PTT-XPP models (a priori containing the GS derivative term) are presented.

Predictive/fitting capabilities of differential constitutive models for polymer melts—reduction of nonlinear parameters in the eXtended Pom-Pom model

A phenomenological modification of the eXtended Pom-Pom (XPP) model is proposed with the aim to reduce the number of free nonlinear parameters. The modified XPP model includes three parameters per mode in total (two linear viscoelastic parameters—linear relaxation time λ and shear modulus G, and one nonlinear parameter). The original XPP model contains five parameters (two linear viscoelastic parameters and three nonlinear ones, one nonlinear parameter participates in the second normal stress difference prediction). The predictive/fitting capabilities of the modified model are compared with the Giesekus, eXtended Pom-Pom, and modified Leonov models using various low-density PE materials in steady and transient shear and uniaxial elongational flows. It has been found that the modified model is capable of predicting/fitting the rheological properties, with the exception of the second normal stress difference, for studied LDPE materials with sufficient accuracy, including strain hardening in uniaxial elongational flow.

Numerical Predictions of Viscoelastic Fluid in Pipe Flow Using Extended Pom-Pom Model

In this study prediction of the steady-state flow of branched polymer melts in pipe geometry with finite volume method is presented. The fluid is modelled using a modification of the Pom-Pom model known as the single eXtended Pom-Pom (XPP) where viscoelastic fluid, is typically a commercial low-density polyethylene. In finite volume method, the operator-integration is used to discretize the governing equations in space or control volume. An iterative solution algorithm that decouples the computation of momentum from that of stress is used to solve the discrete equations. Numerical results are presented; include the profiles of all relevant stresses, the axial velocity, stretch and the viscosity across the gap, demonstrating the performance of the model predictions. The influence of the model parameters on the solution and flow behaviour is described and, in particular, the dependence of velocity and stresses distribution as a function of Weissenberg number is analyzed. Finally, for rectitude and accuracy, the verification of this model is performed with the Gieskus model that shows good conformity between the results.

Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows

AbstractIn this paper we present a finite difference method for solving two-dimensional viscoelastic unsteady free surface flows governed by the single equation version of the eXtended Pom-Pom (XPP) model. The momentum equations are solved by a projection method which uncouples the velocity and pressure fields. We are interested in low Reynolds number flows and, to enhance the stability of the numerical method, an implicit technique for computing the pressure condition on the free surface is employed. This strategy is invoked to solve the governing equations within a Marker-and-Cell type approach while simultaneously calculating the correct normal stress condition on the free surface. The numerical code is validated by performing mesh refinement on a two-dimensional channel flow. Numerical results include an investigation of the influence of the parameters of the XPP equation on the extrudate swelling ratio and the simulation of the Barus effect for XPP fluids.

Effect of Elasticity Parameter on Viscoelastic Fluid in Pipe Flow Using Extended Pom-Pom Model

In this study prediction of the steady-state flow of branched polymer melts in pipe geometry with finite volume method is presented. Our analysis in this study revealed that; for normal-stress τθθ , the XPP model can predict this tensor unlike the other viscoelastic models such as PTT or Gieskus which can not predict τθθ for viscoelastic fluid in two dimensional pipe flows. The fluid is modelled using a modification of the Pom-Pom model known as the single eXtended Pom-Pom (XPP) where viscoelastic fluid is typically a commercial low-density polyethylene. In finite volume method, the operator-integration is used to discretize the governing equations in space or control volume. An iterative solution algorithm that decouples the computation of momentum from that of stress is used to solve the discrete equations. Numerical results are presented, including the profiles of all relevant stresses, the axial velocity, stretch and the viscosity across the gap, demonstrating the performance of the model predictions. The influence of elasticity parameter on flow behaviour is studied, which demonstrates in particular, the dependence of velocity and stresses distribution as a function of Weissenberg number is analyzed. Also, the effect of Weissenberg number on pressure gradient has been considered. Finally, verification of the present model was made by comparing to the Generalized Newtonian Fluid (GNF) model.

Anisotropy parameter restrictions for the eXtended Pom-Pom model

A significant step forward in modelling polymer melt rheology has been the introduction of the Pom-Pom constitutive model of McLeish and Larson [T.C.B. McLeish, R.G. Larson, Molecular constitutive equations for a class of branched polymers: the Pom-Pom polymer, J. Rheol. 42 (1) (1998) 81–110]. Various modifications of the Pom-Pom model have been published over the years in order to overcome several inconveniences of the original model. Amongst those modified models, the eXtended Pom-Pom (XPP) model of Verbeeten et. al. [W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens, Differential constitutive equations for polymer melts: the extended Pom-Pom model, J. Rheol. 45 (4) (2001) 823–843] has received quite some attention. However, the XPP model has been criticized for the generation of multiple and unphysical solutions. This paper deals with two issues. First, in the XPP model, anisotropy is implemented in a Giesekus-like manner which is known to result in unphysical solutions for non-linear parameter values α ≥ 0.5 . Hence, we put forward the conjecture that a similar limitation holds for the XPP model. In the present paper, the limits for the anisotropy parameter are elaborated on and result to be most restraining at high deformation rates where the backbone tube is oriented and the backbone tube stretch approaches the number of arms q. By restricting the anisotropy parameter to a maximum critical value the XPP model produces only one solution, which is the correct physical rheology. In the second part we show that, contrary to the results published by Inkson and Phillips [N.J. Inkson, T.N. Phillips, Unphysical phenomena associated with the extended Pom-Pom model in steady flow, J. Non-Newton. Fluid 145 (2–3) (2007) 92–101], for the special case where the anisotropy parameter equals zero, only one physically relevant solution exists in unaxial extensional. In addition to this physically relevant solution, also solutions exist in the physically unattainable part of the conformation space. However, the existence of these physically unattainable solutions is not a unique feature of the XPP model but rather general for non-linear differential type rheological equations.

Thermodynamic admissibility of the extended Pom-Pom model for branched polymers

The thermodynamic consistency of the eXtended Pom-Pom (XPP) model for branched polymers of Verbeeten et al. [W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens, Differential constitutive equations for polymer melts: the extended pom-pom model, J. Rheol. 45 (4) (2001) 823–843; W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens, Differential constitutive equations for polymer melts: the extended pom-pom model (vol 45, pg 823–843, 2001), J. Rheol. 45 (6) (2001) 1489] as well as its modified version [J. van Meerveld, Note on the thermodynamic consistency of the integral pom-pom model, J. Non-Newtonian Fluid Mech. 108 (1–3) (2002) 291–299] is investigated from the perspective of non-equilibrium thermodynamics, namely the General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) framework. The thermodynamic admissibility of the XPP model is shown for both its original and modified form. According to the GENERIC formalism, the parameter α introduced by Verbeeten et al. to predict non-zero second normal stress in shear flows must fulfill the condition 0 ≤ α ≤ 1.

Stress Induced Crystallization in Elongational Flow

Abstract Stress-induced crystallization is studied in an extensional flow device (a cross-slot flow cell) for an isotactic Polypropylene (iPP) by measuring the micro-structure that develops after flow. Birefringence and Wide Angle X-ray experiments were performed. The birefringence experiments with flow below the melting temperature showed the occurrence of a fiber-like structure around the outflow centerline and, later in time, of ‘streamlines’. The latter are explained by the influence of shear gradients close to the optical windows. The WAXS experiments also showed the fiber-like structure around the outflow centerline, having orientations in the (110), (040) and (130) reflections, with a width of about 80 μm. This dominance of the elongational flow around the stagnation line could not be observed in experiments with flow above and subsequent crystallization below the melting temperature. Structure development in this cell was numerically predicted using the Leonov and the extended Pompom (XPP) model. The oriented structures can be predicted. Moreover, process conditions are found with less influence of the (unwanted) shear gradients. Finally, it was concluded that the strain hardening behavior in the Leonov model over-estimates the first component of the Finger tensor, and thus the number of flow-induced nuclei. Therefore it is recommended to use the extended Pompom model as a more realistic basis for a quantitative flow-induced crystallization model.